Quasi-explicit time-integration schemes for dynamic fracture with set-valued cohesive zone models

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Quasi-explicit time-integration schemes for dynamic fracture with set-valued cohesive zone models

David Doyen, Alexandre Ern, Serge Piperno

To cite this version:

David Doyen, Alexandre Ern, Serge Piperno. Quasi-explicit time-integration schemes for dynamic fracture with set-valued cohesive zone models. Computational Mechanics, Springer Verlag, 2013, 52 (2), pp.401-416. �10.1007/s00466-012-0819-2�. �hal-00736779�

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(will be inserted by the editor)

Quasi-explicit time-integration schemes for dynamic fracture with set-valued cohesive zone models

D. Doyen · A. Ern · S. Piperno

Received: date / Accepted: date

Abstract We investigate quasi-explicit time-integration schemes for solving dynamic fracture problems with set-valued cohesive zone models. These schemes combine a central difference time-integration scheme and a partially implicit and lumped treatment of the cohesive forces. At each time step, the displacements of the nodes in the interior of the domain are computed in an explicit way, while the displacements of each node at the interface are computed by solving a local nonlinear problem. The method provides a general and robust way of treating the set-valued cohesive zone model while keeping a moderate computational cost.

Keywords cohesive zone model·finite elements·time-integration scheme

1 Introduction

Cohesive zone models have been introduced in the late 50s [1, 2, 11]. They can be applied to a large range of materials (concrete, steel, etc...) and fracture processes (brittle fracture, ductile fracture, fatigue, dynamic fracture) and they can be easily enriched with more complex physical behaviors (contact and friction after decohesion, corrosion, etc...). Moreover, cohesive zone models fit quite well within the

This work was partly supported by Electricit´e de France (EdF R&D) through a CIFRE PhD fellowship.

D. Doyen

Université Paris-Est, Laboratoire d’Analyse et de Mathématiques Appliquées (UMR 8050), UPEMLV, UPEC, CNRS, F-77454, Marne-la-Vallée, France

E-mail: david.doyen@univ-mlv.fr A. Ern

Universit´e Paris-Est, CERMICS, Ecole des Ponts ParisTech, 77455 Marne la Vall´ee Cedex 2, France

E-mail: ern@cermics.enpc.fr S. Piperno

Universit´e Paris-Est, CERMICS, Ecole des Ponts ParisTech, 77455 Marne la Vall´ee Cedex 2, France

E-mail: serge.piperno@enpc.fr

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framework of finite elements. For all these reasons, they are now widely used in engineering simulations. A cohesive zone model can describe the mechanical forces along a fracture — it can be simply viewed as a boundary condition. The interface forces depend at least on the opening (displacement jump at the interface). In a typical cohesive zone model, the separation occurs at the interface only after a critical stress has been reached. When the separation has occurred, cohesive forces remain. These forces decrease when the opening increases and tend to van- ish (softening behavior). Physically, the cohesive forces represent the weakening of the material in the fracture process zone ahead of the crack tip. Furthermore, real cracks cannot experience self-healing in general. To take into account this ir- reversibility, one can introduce a history parameter, such as the maximal opening.

For quasi-static fracture, cohesive zone models depending on opening and maximal opening are well established and in good agreement with experiments. In the dynamic case, numerical simulations with such cohesive zone models predict of- ten crack speeds far higher than those observed in the experiments. For instance, for mode-I fracture in brittle materials, numerical crack tip speeds are close to the Rayleigh wave speed, while experimental crack tip speeds nearly reach half of this value (see for instance [22, 20]). To remedy this, cohesive zone models depending on the opening rate have been designed [21, 27]. Such models are called rate-dependent.

Since the crack tip speed is high, typically of the same order as the wave speeds, small time steps are needed to capture accurately the fracture phenomenon.

Therefore, it seems natural to consider an explicit time-integration scheme. For cohesive zone models in which the interface forces are related to the opening by a classical function, the use of an explicit time-integration scheme is straightforward [26]. However, in most cohesive zone models, the interface forces are not related to the opening by a classical function, but by a set-valued map. Indeed, most cohesive zone models feature perfect initial adhesion, contact or rigid unloading.

There are two main difficulties in using fully explicit schemes in such a context.

Firstly, the interface forces are not defined for negative normal opening (see Figure 1, left). Secondly, the interface tangential forces are discontinuous with respect to the tangential opening (see Figure 1, right), and this can cause oscillations. A first option consists in regularizing the set-valued map to turn it into a single- valued map. Unfortunately, the regularization of a non-interpenetration condition deteriorates substantially the stability condition of explicit schemes (the penalty contact condition introduces in the model an artificial stiffness larger than the material stiffness). Moreover, replacing a discontinuity by a very stiff slope does not really solve the problem. Alternatively,ad hocprocedures have also been developed to treat a few specific cases of set-valued interface forces: allowing the separation only after a failure criterion has been reached [5, 19, 12],a posteriorienforcement of the non-interpenetration condition [5, 12], tolerance parameter on the tangential opening [12]. For complex cohesive zone models with several set-valued parts, the combined use of these procedures generally becomes quite intricate, or even unfeasible.

In the present work, we focus on dynamic fracture models where the material can only crack along a prescribed surface (fracture interface). In other words, the crack path is known in advance. This assumption may appear as a limitation.

However, fracture models predicting the crack path are still quite challenging for in- dustrial applications. Moreover, for a large range of applications (interfacial crack,

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small propagation crack), postulatinga priorithe crack path is reasonable (see [16]

for further discussion). We assume that the bulk behavior is governed by the linear elastodynamic equations and that the separation process at the fracture interface obeys a cohesive zone model. We consider a generic cohesive zone model depending on the opening, the opening rate, and the maximal opening. This generic model encompasses most of the usual cohesive zone models. Space semi-discretization is achieved using P1 finite elements. We propose time-integration schemes that combine a central difference scheme with a partially or fully implicit treatment of the interface forces. The central difference scheme is a standard scheme for elastodynamics [15]. The implicit treatment of the interface forces provides a general and robust way of treating the set-valued cohesive zone model. In order to keep a moderate computational cost, we use lumping techniques for the mass term and the interface forces. We thus obtain quasi-explicit methods: at each time step, the displacements of the nodes in the interior of the domain are computed in an explicit way, while the displacements of each node at the interface are computed by solving a small nonlinear problem (this can generally be achieved in an analytical way). First, we consider a fully implicit treatment of the interface forces. However, staggering in time the force at the fracture interface can have a sizeable effect on the energy behavior and the accuracy of the time-integration scheme. Conse- quently, we propose a second time-integration scheme, in which the interface forces are split into a set-valued monotone part and a single-valued softening part. The former is treated in an implicit way, the latter in an explicit way. This improves the accuracy and the energy behavior. Note that some of the ad hoc procedures described above can be loosely interpreted as an implicit treatment of the set valued part of the cohesive zone model (failure criterion,a posteriori enforcement of the contact condition).

We begin by presenting the generic cohesive zone model and examples which fit into this framework (Section 2). We then formulate the continuous problem of dynamic fracture (Section 3). Sections 4 and 5 are devoted to the finite element discretization in space and to the time-integration schemes, respectively. We dis- cuss the implementation of the schemes in Section 6. Finally, numerical results are presented in Section 7, and conclusions are drawn in Section 8.

2 Cohesive zone model

We consider a generic cohesive zone model. The forces at the fracture interface are described by a set-valued map which depends on the opening, the opening rate, and the maximal effective opening (the notion of effective opening is defined below).

2.1 Generic model

Let (·,·) denote the usual scalar product in Rⁿ (n≥ 1) and let | · | denote the corresponding Euclidean norm. LetP(Rⁿ) denote the set of all subsets ofRⁿ. In a d-dimensional problem (d= 2 ord= 3), the cohesive zone model is characterized by a set-valued mapR:R^d×R^d×R^d→ P(R^d). The arguments ofRare the maximal effective opening, the opening rate, and the opening, respectively. For each

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triplet (δ, z, p), the mapR(δ, z, p) yields a set of vectors, which are the admissible interaction forces. The first component ofλ∈R(δ, z, p) is the normal force at the interface and the second and third ones are the tangential forces. For an opening p ∈ R³ (resp. p ∈ R²), the effective opening is defined as ¯δ(p) = (p1,|p2|,|p3|) (resp. ¯δ(p) = (p1,|p₂|)).

Since a cohesive zone model describes a softening behavior, the set-valued map Ris not monotone with respect to the opening. However, the slope of the softening part ofR is assumed to be bounded. This assumption is, in particular, useful to establish the well-posedness of our first time-integration scheme (see Proposition 1).

Assumption 1 The operatorRsatisfies the following one-sided Lipschitz condition:

there is a real numberCLsuch that, for allδ∈R^d, for allz∈R^d, for allp,q∈R^d, (λp−λq, p−q)≥ −CL|p−q|², ∀λp∈R(δ, z, p), ∀λq∈R(δ, z, q). (1) The present generic model encompasses for instance the Camacho-Ortiz law [5] and the Talon-Curnier law [23], but not the rectangular law (because of the infinite slope of the softening part).

Remark 1 In most cohesive zone models, the operatorR(δ, z,·) is built as the dif- ferential (in a generalized sense) of an energy. This operator being non-monotone, the associated cohesive energy is non-convex.

Remark 2 After decohesion, contact closure can lead to friction phenomena at the interface, which can play an important role in the fracture process. Cohesive zone models including friction have been proposed for instance in [18]. The friction force is generally a monotone set-valued function of the tangential velocity at the interface and could be easily added to our generic model.

2.2 Examples

This section collects some examples of cohesive zone models fitting the above framework. The first two examples can be viewed as simplified variants of the Camacho-Ortiz law [5].

A reversible triangular model with uncoupled normal and tangential forces This model depends on the opening and prescribes uncoupled normal and tangential interface forces. It relies on two parameters:σc, the maximal cohesive force, anddc, the critical opening. It can be represented by a set-valued map R:R^d → P(R^d) whose components are independent. The normal component R1 : R → P(R) is such that

R1(p) :=











(−∞, σc] ifp= 0, σc

1−_d^p

c

if 0< p≤dc, 0 ifdc< p,

∅ ifp <0.

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dc

σc

λ1

p1

λ2

p2

−σc

dc

σc

Fig. 1 Triangular model with uncoupled normal and tangential forces. Normal force (left).

Tangential force (right).

For simplicity, in the definition of cohesive zone models, a singleton{x}is simply denoted x. The tangential components R2 :R → P(R) and R3 : R→ P(R) are such that

R2(p) =R3(p) :=











0 ifp <−d_c,

−σc

1 +_d^p

c

if −dc≤p <0, [−σc, σc] ifp= 0,

σc

1− _d^p

c

if 0< p≤dc, 0 ifdc< p.

This model is represented in Figure 1. It is easy to check that this model satisfies Assumption 1 with CL = σc/dc. Moreover, energies Ψ1 : R⁺ →R, Ψ2 :R →R, andΨ3:R→Rcan be associated with this model, namely

Ψ1(p) :=

(σcp 1−_2d^p

c

if 0≤p≤dc,

1

2σcdc ifdc< p, and

Ψ2(p) =Ψ3(p) :=











1

2σcdc ifp <−d_c,

−σcp

1 +_2d^p

c

if −dc≤p <0, σcp

1−_2d^p

c

if 0≤p≤dc,

1

2σcdc ifdc< p.

An irreversible triangular model with only normal force This model depends on the normal openingpand maximal effective normal openingδ and prescribes only the normal force. Moreover, it is irreversible with a linear unloading. As the previous model, it involves two parameters: σc, the maximal cohesive force, and dc, the critical opening. It can be represented by the set-valued mapR^irr₁ :R×R→ P(R)

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dc

σc

λ

δ p dc

σc

λ

δ p

Fig. 2 An irreversible triangular model with only normal force. Linear unloading (left). Rigid unloading (right).

such that

R^irr1 (δ, p) :=











(−∞, σc] ifδ=p= 0, (−∞,0] if 0 =p < δ, σc

1−_d^δ

c

p

δ if 0< p≤δ≤dc, σc

1−_d^p

c

if 0< δ < p≤dc, 0 ifdc< p, 0≤δ, 0 ifdc< δ,0≤p,

∅ otherwise.

This model is represented in Figure 2, left. It satisfies Assumption 1 withCL = σc/dc. A common variant of this model consists in replacing the linear unloading with a rigid unloading (Figure 2, right). An energy Ψ₁^irr : R⁺×R⁺ →R and a dissipated energy ˜Ψ₁^irr : R⁺ → R can be associated with the irreversible model with linear unloading. They are defined as follows:

Ψ1^irr(δ, p) :=









 σc

−^δ₂+p− _2d^p²

c

if 0< δ < p≤dc, σc

1−_d^δ

c

p²

2δ if 0< p≤δ≤dc,

1

2σcdc ifdc< p, δ= 0, 0 ifdc< δ,0≤p, Ψ˜1^irr(δ) :=

(₁

2σcδ if 0≤δ≤dc,

1

2σcdc ifdc< δ.

A rate-dependent triangular model with only normal force [27] This model depends on the normal openingpand the normal opening ratez, and prescribes only the normal interface force. It relies on three parameters: σc, the maximal cohesive force,dc, the critical opening, andη, a viscosity parameter. It can be represented by the set-valued mapR₁^visc:R×R→ P(R) such that

R^visc1 (z, p) :=











(−∞, σc] ifp= 0, σc

1−_d ^p

c(1+ηz⁺)

if 0< p < dc(1 +ηz⁺),

0 ifdc(1 +ηz⁺)≤p,

∅ otherwise,

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λ

dc d˜c p σc

Fig. 3 A rate-dependent triangular model with only normal force ( ˜dc:=dc(1 +ηz⁺)).

wherez⁺ denotes the positive part ofz. This model is represented in Figure 3. It satisfies Assumption 1 withCL =σc/dc.

2.3 Link with Griffith’s model

It is possible to make a link between cohesive zone models and Griffith’s models.

When cohesive forces act over a sufficiently short range, the stress fields near the crack tip are equivalent in both models. Furthermore, the material parameter used in Griffith’s model, the fracture toughness Gc, is equal to the energy needed to completely open the crack in the cohesive zone model. A formal argument for this asymptotic analysis can be found in [25] and a rigorous proof for a simple model in [17]. The fracture toughness corresponding to the rate-independent triangular models (presented above) is

Gc= 1 2σcdc.

In the rate-dependent triangular model, the fracture toughness increases with the opening rate.

3 Continuous problem

We now formulate the governing equations of the dynamic fracture problem.

3.1 Geometry

We consider a domain Ω ⊂ R^d (d= 2 or d= 3) and we assume that the crack can only appear on a (d−1)-dimensional smooth surface Γ (see Figure 4). We call Γ the fracture interface. We set ˜Ω := Ω\Γ. We can fix an orientation and define two sides for Γ, a positive side and a negative side. Let v : Ω → R^d be a displacement field. The trace of v on the positive side is denotedv⁺, the trace on the negative side is denoted v⁻. We denote ν the unit normal vector to Γ pointing to the positive side. We define two tangential unit vectors τ1 andτ2, so that (ν, τ1, τ2) forms a local direct orthonormal basis (obviously, ford= 2, only

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Γ^D Γ^N

Ω˜ − +

Γ ν

Fig. 4 Geometric setup.

one tangential vector is considered). The displacement jump ofv at the interface is defined as

[[v]] =v⁺−v⁻. (2)

To define the interface forces, we take into account the local orientation of the interface by introducing the rotation matrixQtransforming the canonical basis of R^d into (ν, τ1, τ2).

3.2 Governing equations

The material is supposed to be linear isotropic elastic with Young modulus E, Poisson ratio νP, and mass density ρ. The elasticity tensor is denoted A. An external loadf is applied to the body. Letu:Ω×(0, T)→R^d,(u) :Ω×(0, T)→ R^d,d, andσ(u) :Ω×(0, T)→R^d,dbe the displacement field, the linearized strain tensor, and the stress tensor, respectively. Denoting time-derivatives by dots, the momentum conservation equation reads

ρ¨u−divσ=f, σ=A:, = 1

2(∇u+^T∇u) in ˜Ω×(0, T). (3) The boundary∂ΩofΩis partitioned into two disjoint subsetsΓ^DandΓ^N. Dirich- let and Neumann conditions are prescribed onΓ^D andΓ^N, respectively,

u=uD on Γ^D×(0, T), σ·ν=fN onΓ^N×(0, T). (4) OnΓ, the cohesive law is enforced

σ(u⁻)·ν=−σ(u⁺)·ν=:λ, λ∈^TQR(δ, Q[[ ˙u]], Q[[u]]). (5) The maximal effective openingδ is defined, fort >0, by

δ(t) = sup

s∈[0,t)

¯δ(Q[[u(s)]]). (6)

At the initial time, the displacement, the velocity, and the maximal effective opening are prescribed:

u(0) =u⁰, u(0) =˙ v⁰, δ(0) =δ⁰. (7) Equations (3)-(5) can be written in a variational form: seek u such that, for all test functionv,

Z

Ω˜

ρ¨u·v+ Z

Ω˜

(u) :A:(v) = Z

Ω˜

f(t)·v+ Z

Γ^N

fN(t)·v− Z

Γ

λ·[[v]], (8) whereλ∈^TQR(δ, Q[[ ˙u]], Q[[u]]).

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3.3 Mathematical aspects

The mathematical analysis of Problem (3)-(7) is beyond the scope of the present work. However, let us mention some related results.

– In the quasi-static case with a reversible cohesive zone model (with perfect adhesion or not), the existence is proven. The solution is in general not unique [10].

– In the quasi-static case with an irreversible cohesive zone model, the existence of a solution is proven in [8, 4].

– In the dynamic case, it should be possible to prove, using compactness arguments similarly to [9], existence for a visco-elastic material and a reversible cohesive law with perfect adhesion, and even to prove existence and uniqueness for an elastic material and a regularized cohesive law.

3.4 Length and time scales

In order to capture accurately a phenomenon with numerical simulations, it is important to choose a time step and a mesh size which resolve its time and length scales. In a dynamic fracture problem, the relevant length scale is the lengthlcoh

of the cohesive zone (the part of the interface which is not completely cracked and where cohesive forces act). The relevant time scale is the crack tip speed divided by the cohesive zone length. Several methodologies have been proposed in the literature to estimate the cohesive zone length; see [24] and references therein.

They all yield, in the case of plane strain and triangular cohesive zone models, an estimation of the form

lcoh=M E 1−ν_P²

Gc

σ_c², (9)

where M is a parameter close to 1. For an isotropic linear elastic material and a Griffith model of fracture, a theoretical analysis predicts that the limiting crack tip speed for a mode-I fracture is the Rayleigh wave speed [13, 3]. For mode-II and mode-III fractures, the limiting speeds are the dilatational wave speed and the shear wave speed, respectively [13, 3]. The dilatational and shear wave speeds are given by the following formulae:

cd= s

E(1−νP)

ρ(1 +νP)(1−2νP), cs= s

E

2ρ(1 +νP). (10) The Rayleigh wave speed can be estimated by the following expression [13]

cR≈cs0.862 + 1.14νP

1 +νP

. (11)

Rate-dependent cohesive zone models involve an additional time scale, linked to the opening rate and generally smaller than the time scale linked to the crack tip speed. At least in the quasi-static evolution, analytical estimations of the opening time are provided in [7].

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4 Finite element discretization

In this section, we describe the space approximation of the dynamic fracture problem. Linear finite elements are used together with lumping of the mass term and the interface forces.

4.1 Finite element spaces

In 2D (resp. in 3D), the domain Ω is approximated by a polygon (resp. a poly- hedron) Ωh and the interfaceΓ by a polygonal curve (resp. a polygon)Γh. The domainΩh is meshed with triangles (resp. tetrahedra) conforming to the interface Γh. LetT_hdenote the mesh overΩhand letF_hcollect the faces located onΓh. Let Ω˜h=Ωh\Γh. Let{xi}i∈N be the nodes of the meshT_hwhereN collects the node indices. Let N^D be the indices of nodes where a Dirichlet condition is enforced and byN^cthe indices of nodes lying onΓh. The displacements are approximated withP1finite elements:

V_h={v_h∈C⁰( ˜Ωh)^d; vh|T ∈(P¹)^d, ∀T ∈ T_h, andvh(xi) = 0, ∀i∈ N^D}.

Note that functions in Vh can be discontinuous across Γh. We consider the La- grange nodes ofV_h and denote them{ξi}_i∈NLwhereN^L collects the corresponding node indices. The Lagrange nodes are not exactly the mesh nodes {xi}i∈N

because of the discontinuity at the interface. Specifically, for each mesh node xi

lying on Γh, there are two Lagrange nodes ξ_i⁺ and ξ_i−. For all i ∈ N^c, for all vh∈ Vh, we set

vh(ξ_i⁺) :=v⁺_h(xi), vh(ξ_i⁻) :=v⁻_h(xi). (12) The cohesive forces are also approximated byP1finite elements,

L_h={λ_h∈C⁰(Γh)^d; λh|F ∈(P1)^d, ∀F ∈ F_h}.

At each nodexilying onΓh, we define normal and tangential unit vectors (νi, τ1i, τ2i) forming a direct orthonormal basis. LetQi be the associated rotation matrix. We define also the set-valued operatorRi:

Ri(·,·,·) =^TQiR(·, Qi·, Qi·). (13)

4.2 Lumping of the mass term and the cohesive term

The mass term and the cohesive forces term are lumped. Mass lumping is usual with explicit time-integration schemes. It yields an easy-to-invert mass matrix at each time step, while improving the CFL condition [15]. A way of lumping the mass term is to evaluate it with an approximate quadrature whose Gauss points are the nodes of the finite element space. ForP1 finite elements, it is usual to use the following quadrature formula (in dimensiond)

Z

T

f≈

d+1

X

i=1

|T|

d+ 1f(αi), (14)

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where |T| is the measure of the simplex T and {α_i}_1≤i≤d+1 its vertices. This quadrature is second-order accurate. The lumped mass term ˆmh:Vh× Vh→Ris built with this quadrature by setting

ˆ

mh(vh, wh) = X

i∈N^L

µivh(ξi)·wh(ξi), (15) where

µi= X

T∈Ti

ρ|T|/(d+ 1), (16)

T_i being the set of elements for whichξi is a vertex. The lumped cohesive term ˆbh:L_h× V_h→Ris such that

ˆbh(λh, wh) = X

i∈N^c

βiλh(xi)·

v_h⁻(xi)−v⁺_h(xi)

, (17)

where

βi= X

F∈Fi

|F|/d, (18)

F_i being the set of faces for which xi is a vertex and |F| the measure of the faceF. Finally, the stiffness term ah :V_h× V_h →R and the external force term lh: [0, T]× Vh→Rare built in a standard way, namely

ah(vh, wh) :=

Z

Ω˜h

(vh) :A:(wh), (19)

lh(t, vh) :=

Z

Ω˜h

f(t)·vh+ Z

Γ^N

fN(t)·vh, (20) up to quadratures forlh. We define the matrices ˆMh,Kh, and ˆBhassociated with the bilinear forms ˆmh, ah, and ˆbh, respectively. We also define Lh(t) to be the column vector associated with the linear formlh(t,·). Foruh ∈ Vh, we define Uh

as the column vector whose components are the coordinates of uh in the finite element basis. We denote NV the size of Uh. We denote Uh,i the d-dimensional sub-vector associated with the Lagrange nodeξi. Similarly, forλh∈ L_h, we define Λh as the column vector whose components are the coordinates ofλh in the finite element basis. We denote NΛ the size of Λh. We denote Λh,i the d-dimensional sub-vector associated with the nodexi. Finally, we define, for alli∈ N^c,

[[Uh]]i =U_h,i⁺−U_h,i⁻ and {U_h}_i= U_h,i⁺+U_h,i⁻

2 . (21)

For eachi∈ N^L, we denote respectively ˆMh,i andKh,i thed×dsub-matrices of Mˆh andKh associated with the Lagrange node ξi. For eachi ∈ N^c, we denote Bˆh,i thed×dsub-matrix of ˆBh associated with the nodexi. We define the set- valued operatorRh :R^N^Λ×R^N^Λ×R^N^Λ → P(R^N^Λ) such that for all∆h ∈R^N^Λ, Zh∈R^N^Λ,Ph∈R^N^Λ,

Λh∈Rh(∆h, Zh, Ph)⇐⇒Λh,i∈Ri(∆h,i, Zh,i, Ph,i)∀i∈ N^c. (22)

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The space semi-discrete problem takes the form

MˆhU¨h(t) +KhUh(t) =Lh(t) + ˆBhΛh(t), (23) Λh(t)∈Rh(∆h(t),[[ ˙Uh(t)]],[[Uh(t)]]), (24)

∆h,i(t) = sup

s∈[0,t)

δ(Q¯ i[[Uh(s)]]i), ∀i∈ N^c. (25) Remark 3 In the present work, we consider P1 finite elements. Other types of finite elements can be used, provided an accurate lumping technique is available.

For instance, this is the case forQ1 elements (see [15]). ForPk andQk elements with k ≥ 2, the lumping techniques are more subtle (see for instance [6] and references therein).

Remark 4 Because of lumping, even when the nodes of the interface are in the perfect adhesion regime, our discretization is not equivalent to a discretization without interface. Consequently, small spurious wave reflections can occur at the interface in the perfect adhesion regime.

5 Time-integration schemes

It remains now to discretize the problem in time. The time-integration schemes we propose are based on the central difference scheme. To begin with, let us recall the main properties of this scheme in the linear elastodynamic case. We then describe and analyze two schemes for the dynamic fracture problem.

5.1 Central differences for elastodynamics

For simplicity, the interval [0, T] is divided into equal subintervals of length ∆t.

We set tⁿ = n∆t and denote U_hⁿ the approximation of Uh at time tⁿ. For the central difference scheme, the discrete velocity and the discrete acceleration are defined respectively as

U˙hⁿ:= U_hⁿ⁺¹−U_hⁿ⁻¹

2∆t and U¨hⁿ:= U_hⁿ⁺¹−2U_hⁿ+U_hⁿ⁻¹

∆t² . (26)

At each time step of the central difference scheme, one seeksU_hⁿ⁺¹ such that 1

∆t²Mˆh(U_hⁿ⁺¹−2U_hⁿ+U_hⁿ⁻¹) +KhU_hⁿ=Lh(tⁿ). (27) The central difference scheme exhibits a stability condition (CFL condition) of the form

cd∆t≤O(hmin), (28)

where hmin is the smallest mesh element diameter. An admissible value of the constant in the CFL condition can be specified in 1D and for structured meshes in higher dimension. The elastic energy, the kinetic energy, and the total energy of the discrete system at timetⁿare respectively defined by

E_elⁿ := 1

2(KhU_hⁿ, U_hⁿ), E_kinⁿ := 1 2

MˆhU˙_hⁿ,U˙_hⁿ

, Eⁿ:=E_elⁿ+Eⁿ_kin. (29)

(14)

In linear elastodynamics with no forcing termLh(tⁿ), the central difference scheme does not preserve the energy. Nevertheless, the scheme preserves the following quadratic form, referred to as a shifted energy,

Eⁿ:=Eⁿ−∆t² 8

MˆhU¨_hⁿ,U¨_hⁿ

. (30)

With a forcing term, the following shifted energy balance holds true:

Eⁿ⁺¹− Eⁿ= 1

2(Lh(tⁿ⁺¹) +Lh(tⁿ), U_hⁿ⁺¹−U_hⁿ). (31)

5.2 Scheme A (fully implicit interface forces)

The scheme A combines a central difference scheme with an implicit treatment of the interface forces. More precisely, the interface forces are implicit in the opening, while they are explicit in the opening rate and in the maximal effective opening.

Scheme A.SeekU_hⁿ⁺¹∈R^N^V andΛⁿ⁺¹_h ∈R^N^Λ such that

1

∆t²

Mˆh(U_hⁿ⁺¹−2Uhⁿ+U_hⁿ⁻¹) +KhUhⁿ=Lh(tⁿ) + ˆBhΛⁿ⁺¹_h , (32) Λⁿ⁺¹_h ∈Rh(∆ⁿh, Zhⁿ,[[U_hⁿ⁺¹]]), (33) where, for alli∈ N^c,

Z_h,iⁿ = [[U_hⁿ]]i−[[U_hⁿ⁻¹]]i

∆t , ∆ⁿ_h,i= max(∆ⁿ⁻¹_h,i ,δ(Q¯ i[[U_hⁿ]]i)). (34) A way of implementing this scheme will be described in Section 6.1. We now prove that, at each time step, the problem is well-posed under a mild restriction on the time step. We observe that condition (36) below is indeed mild, since for hcandhmin of the same size and small enough, the stability condition (28) of the central difference scheme imposes a more stringent limit on the time step than the well-posedness condition (36).

Proposition 1 (Well-posedness)Problem (32)-(34) has a unique solution under the conditions

µ_i⁺

∆t² >2CLβi and µ_i−

∆t² >2CLβi, ∀i∈ N^c, (35) where the coefficientsµi andβi are defined by (16) and (18). For a quasi-uniform mesh, the above condition can be rewritten as

∆t² hc

< C, (36)

where hc is the mesh size at the interface and C a constant independent of the mesh size.